the data set represents the total number of pencils each student in a class needs to sharpen. 0, 1, 1, 1, 2…

the data set represents the total number of pencils each student in a class needs to sharpen. 0, 1, 1, 1, 2, 3, 4, 4, 6, 6, 9. which box - plot correctly represents the data?

the data set represents the total number of pencils each student in a class needs to sharpen. 0, 1, 1, 1, 2, 3, 4, 4, 6, 6, 9. which box - plot correctly represents the data?

Answer

Answer:

To determine the correct box - plot, we first need to find the five - number summary (minimum, first quartile $Q_1$, median, third quartile $Q_3$, maximum) of the data set $0,1,1,1,2,3,4,4,6,6,9$.

  1. Find the minimum and maximum:
    • The minimum value of the data set is $0$.
    • The maximum value of the data set is $9$.
  2. Find the median ($Q_2$):
    • There are $n = 11$ data points. The median is the $\left(\frac{n + 1}{2}\right)$-th value.
    • $\frac{11+1}{2}=6$ - th value. So, the median $Q_2=3$.
  3. Find the first quartile ($Q_1$):
    • The lower half of the data set is $0,1,1,1,2$. There are $n_1 = 5$ data points in the lower half. The first quartile is the $\left(\frac{5 + 1}{2}\right)$-th value of the lower - half data set.
    • $\frac{5+1}{2}=3$ - th value. So, $Q_1 = 1$.
  4. Find the third quartile ($Q_3$):
    • The upper half of the data set is $4,4,6,6,9$. There are $n_2 = 5$ data points in the upper half. The third quartile is the $\left(\frac{5+1}{2}\right)$-th value of the upper - half data set.
    • $\frac{5 + 1}{2}=3$ - th value. So, $Q_3=6$.

The box - plot should have a minimum at $0$, $Q_1$ at $1$, median at $3$, $Q_3$ at $6$, and maximum at $9$.

We would need to visually inspect the given box - plots to determine which one has these correct values for the minimum, $Q_1$, median, $Q_3$, and maximum. Without seeing the full set of box - plots clearly, we can't give a definite choice among the options, but the correct box - plot should have the box starting at $1$ (for $Q_1$), ending at $6$ (for $Q_3$), a line inside the box at $3$ (for the median), and whiskers extending to $0$ (minimum) and $9$ (maximum).

Explanation:

Step1: Identify minimum and maximum

Minimum = $0$, Maximum = $9$

Step2: Calculate the median

$n = 11$, $\text{Median}=\text{6 - th value}=3$

Step3: Calculate first quartile

Lower - half data: $0,1,1,1,2$, $n_1 = 5$, $Q_1=\text{3 - th value}=1$

Step4: Calculate third quartile

Upper - half data: $4,4,6,6,9$, $n_2 = 5$, $Q_3=\text{3 - th value}=6$